Extensions with Abelian Kernels in Protomodular Categories

As observed by J. Beck, and as we know from M. Barr's and his joint work on triple cohomology, the classical isomorphism Opext ≅ 𝐻2 that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The same is known for many other “group-like” algebraic structures, and now we present a purely-categorical version of that equivalence, essentially by showing that all torsors are extensions with abelian kernels in any pointed protomodular category, and by giving a necessary and sufficient condition for the converse.